发表机构
School of Science, Xihua University; School of Mathematics, Guangxi University; School of Mathematics and Statistics, Shandong University of Technology; School of Mathematics and Statistics, Henan Normal University（西华大学理学院; 广西大学数学学院; 山东理工大学数学与统计学院; 河南师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了当 $n=pq$（$7\le p<q$ 为素数）时，旋转木马锦标赛 $T_n$ 存在循环哈密顿圈分解，为凯利猜想的保对称扩展提供了无限族支持。
AI 中文摘要
凯利猜想询问每个正则锦标赛是否允许哈密顿圈分解。受其保对称扩展的启发，我们研究旋转木马锦标赛的循环哈密顿分解。对于奇数 $n$，设 $T_n=\Cay\left( \mathbb Z_n,\left\{1,2,\ldots,\frac{n-1}{2}\right\} \right)$ 为旋转木马锦标赛。我们询问 $T_n$ 是否具有在平移 $1$ 下不变的哈密顿分解。虽然当 $n$ 为素数时答案是直接的，但合数阶引入了真正的障碍：非单位差生成短圈而非哈密顿圈。我们通过证明当 $n=pq$ 且 $7\le p<q$ 为素数时，锦标赛 $T_n$ 允许循环哈密顿圈分解，解决了一般的合数阶族。证明结合了素数域上的哈密顿差分序列与一个匹配论证，该论证构造了在 $\mathbb Z_{pq}$ 中具有不相交差集的两条基路径。因此，我们的结果给出了一个无限族，支持正则锦标赛哈密顿分解问题的循环保对称扩展。
英文摘要
Kelly's conjecture asks whether every regular tournament admits a Hamilton cycle decomposition. Motivated by its symmetry-preserving extension, we study cyclic Hamilton decompositions of carousel tournaments. For an odd integer $n$, let \[ T_n=\Cay\left( \mathbb Z_n,\left\{1,2,\ldots,\frac{n-1}{2}\right\} \right) \] be the carousel tournament. We ask whether $T_n$ has a Hamilton decomposition invariant under translation by $1$. Although the answer is immediate when $n$ is prime, composite orders introduce a genuine obstruction: nonunit differences generate short cycles rather than Hamilton cycles. We resolve a general composite-order family by proving that, whenever $n=pq$ for primes $7\le p<q$, the tournament $T_n$ admits a cyclic Hamilton cycle decomposition. The proof combines Hamiltonian difference sequences over prime fields with a matching argument that constructs two base paths with disjoint difference sets in $\mathbb Z_{pq}$. Thus our result gives an infinite family supporting the cyclic, symmetry-preserving extension of the Hamilton decomposition problem for regular tournaments.
Comments17 pages
